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Let's simplify each of the radicals one by one.

### 1. \(\sqrt[3]{\frac{27x^6 y^9}{125x^3 y^5}}\)

- First, simplify the numerical part: \(\frac{27}{125}\). Since both are perfect cubes (\(27 = 3^3\) and \(125 = 5^3\)), we get:
  \[
  \sqrt[3]{\frac{27}{125}} = \frac{3}{5}
  \]
- For the \(x\)-terms, \(\frac{x^6}{x^3} = x^{6-3} = x^3\), and then taking the cube root gives \(x\).
- For the \(y\)-terms, \(\frac{y^9}{y^5} = y^{9-5} = y^4\), and taking the cube root gives \(y^{4/3}\).

Thus, the simplified expression is:
\[
\frac{3x y^{4/3}}{5}
\]

---

### 2. \(\sqrt{64m^3n^4}\)

- Simplify the square root of 64: \(\sqrt{64} = 8\).
- For the \(m\)-terms: \(\sqrt{m^3} = m^{3/2} = m\sqrt{m}\).
- For the \(n\)-terms: \(\sqrt{n^4} = n^2\).

Thus, the simplified expression is:
\[
8n^2 m\sqrt{m}
\]

---

### 3. \(\sqrt{20a^{10}b^{20}c^9}\)

- Simplify the numerical part: \(\sqrt{20} = 2\sqrt{5}\).
- For the \(a\)-terms: \(\sqrt{a^{10}} = a^5\).


Simplify the following radicals:
1. ∛(27x⁶y⁹ / 125x³y⁵)
2. √(64m³n⁴)
3. √(20a¹⁰b²⁰c⁹)
4. ⁵√(486x¹⁷y¹³ / 2050z¹⁸) · 2(⁴√(16x⁸ / y¹⁶z²⁰))
5. ³√(680m⁸ / n¹¹o¹⁶) · ³√(270m¹² / m¹¹) · 4m³n⁴

Learn how to simplify complex radical expressions with exponents and variables in this detailed guide. We cover the properties of radicals, laws of exponents, and provide solutions for problems involving cube roots, square roots, and higher-order roots. Suitable for high school students in Grades 10-12.

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